Cooper instability and superconductivity of the Penrose lattice

Bulk superconductivity (SC) has recently been observed in the Al-Zn-Mg quasicrystal (QC). To settle several fundamental issues of the SC on the QC, we use an attractive Hubbard model to perform a systematic study on the Penrose lattice. The first issue is the Cooper instability of the QC, i.e., no Fermi surface under an infinitesimal attractive interaction. Starting from the two-electron problem outside a filled Fermi sea, we analytically prove that an infinitesimal Hubbard attraction can lead to the Cooper instability as long as the density of the state is nonzero at the Fermi level. The findings provide a basis for the SC on the QC. Our numerical results show that the Cooper pairing always takes place between two time-reversal states, satisfying Anderson’s theorem. On this theorem, we perform a mean-field (MF) study at zero and finite temperatures. The MF study shows that an arbitrarily weak attraction can lead to a pairing order, with the resulting pairing state being well described by the Bardeen-Cooper-Schrieffer theory and the thermal dynamic behaviors being well consistent with the experimental results. The second issue is about the superfluid density on the QC without translational symmetry. Our findings clarify that although the normal state of the system locates at the critical point of the metal-insulator transition, the pairing state exhibits a real SC, carrying finite superfluid density that can be verified by the Meissner effect. This finding is consistent with the experiment results. This study also reveals that the properties of the SC on the Penrose lattice are universal for all QCs.